Solving Three Dimensions Volterra Integral Equations (TDVIE) via a Neural Network
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Abstract
The aim of this paper is present a new numerical method for solvingThree Dimensions Volterra Integral Equations using artificial neural network by design multilayer feed forward Neural Network. A multi- layers design in our proposed method consist of a hidden layer having seven hidden units. and one linear output unit. Linear Transfer function used as each unit and using Levenberg- Marquardtalgorithmtraining. Moreover, examples on three- dimensional Volterra integral equations carried out to illustrate the accuracy and the efficiency of the presented method. In addition, some comparisons among proposed method and Shifted Chebyshev Polynomials method and Reduced Differential Transform Method are presented
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References
1. Atkinson, K.E. (1997). The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press.
2. Chari M. V. K. and Salon S. J. (2000); Numerical Methods in Electromagnetism. Academic Press.
3. Borhanifar A. and Sadri K. (2014); Numerical solution for system of two- dimensional integral equation by using Jacobi operational collocation method. Sohag J. Math. 1, No. 1, , 15-26.
4. Hendi F. A. and Bakodah H.O. (2011); Discrete Adomian Decomposition Solution of Non- linear Mixed Integral Equation. Journal of American Science,7 (12),1081- 1084
5. Mirzaee F. and Hadadiyan E. (2012),; Approximate solutions for mixed nonlinear Volterra- Fredholm type integral equations via modified block- pulse functions. Journal of the Association of Arab Universities for Basic and Applied Science12, 65- 73.
6. Ziyaee F., Tari A.(2014); Regularization Method for Two- dimensional Fredholm Integral Equations of the First Kind. International Journal of Nonlinear Science. Vol.18 No.3, 189- 194.
7. Saberi Nadjagi J., Reza NavidSamadi O. and Tohid E. I (2011); Numerical Solution of two- dimensional Volterra Integral Equations by Spectral Galerkin Method. Journal of Applied Mathematics and Bioinformatics, vol. 1, no. 2, , 159- 174.
8. Abdou M.A., Badr A.A. and Soliman M.B. (2011); On a method for solving a two- dimensional non-linear integral equation of the second kind. J. Comput. Appl. Math., 235, 35893598
9. Saeed R. K. and Berdawood K. A. (2016),; Solving Two- dimensional Linear Volterra- Fredholm Integral Equations of Second Kind by Using Successive Approximation Method and method of Successive Substitutions. ZANCO Journal of Pure and Applied Sciences. 28 (2); 35-46.
10. Sadigh Behzadi Sh. (2012); The Use of Iterative Methods to Solve Two - Dimensional Nonlinear Volterra – FredholmIntegro - Differential Equations. Communication in Numerical Analysis. 1-20.
11. AVAZZADEH Z. and HEYDARI M. (2012); Chebyshev polynomials for solving two-dimensional linear and nonlinear integral equations of the second kind. Computational and applied Mathematics. Volume 31, No.1, 127- 142.
12. Avazzadeha Z., Heydarib M. and Loghmania G. B. (2011); A Comparison between Solving Two- Dimensional Integral Equations by the Traditional Collocation Method and Radial Basis Function. Applied Mathematics Sciences, Vol. 5, no. 23, 1145-1152.
13. Basseem(2015); Degenerate Kernel Method for Three Dimension Nonlinear Integral Equations of the Second Kind. University Journal of Integral Equations 3,61-66.
14. Ziqan A., Armiti S. and I. Suwan(2016); Solving three- dimensional Volterra integral equation by the reduced differential transform method. International journal of Applied Mathematical Research. 5 (2) 103-106.
15. Bakhshi M. and Asghari-Larimi M. (2012); Three- dimensional differential transform method for solving three-dimensional Volterra integral equations. The Journal of Mathematics and Computer Science Vol. 4 No. 2 246-256.
16. Mohamed D. Sh.: Shifted Chebyshev polynomials for Solving Three- Dimensional Volterra Integral Equations of the second kind. https://www.researchgate.net/publication /308692522
17. Hristev R. M. , (1998). The ANN Book.GNU public license, Edition 1 .